About foundamental theorem of algebra, there is a large collection different demonstrations.

I ask: there is some proof that avoid AC (choise axiom)  ?

In a general topos (with natural number object) there are the two construction of real numbers (generalization of the Dedekind and Cauchy classical) that are different.

In ZF theory, are the Dedekind and Cauchy constructions different? (in the "Cauchy" reals, operates on a real number $r$ through a choose of a Cauchy sequence converging to $r$).

About fundamental theorem of algebra, there is a large collection different demonstrations.

I ask: is there some proof that avoids AC (choice axiom)?

In a general topos (with natural number object) there are the two constructions of real numbers (generalizations of the classical Dedekind and Cauchy constructions) that are different.

In ZF theory, are the Dedekind and Cauchy constructions different? (In the "Cauchy" reals, operates on a real number $r$ through a choice of a Cauchy sequence converging to $r$.)